Foundations

Keep the heat and work sign convention explicit in an energy balance.

Apply entropy increase to the whole isolated system.

A simple gas model relates pressure, amount, volume and temperature.

The Boltzmann constant has an exact SI numerical value.

A temperature gradient drives conductive heat flux.

Energy changes phase at the specified coexistence conditions.

Vapor pressure changes with temperature through latent enthalpy.

Thermal equilibrium fixes an ideal absorber's total emitted power.

Fixed-temperature equilibrium weights states by energy.

Microstate probabilities quantify statistical entropy.

Indistinguishable particles have different equilibrium state populations.

For a simple compressible equilibrium system, U(S,V,{N_i}) has differential dU=T dS−p dV+Σ_i μ_i dN_i.

The Legendre transforms H=U+pV and F=U−TS change the natural state variables: dH=T dS+V dp and dF=−S dT−p dV at fixed composition.

Equality of mixed derivatives of smooth equilibrium potentials gives (∂T/∂V)_S=−(∂p/∂S)_V, (∂T/∂p)_S=(∂V/∂S)_p, (∂S/∂V)_T=(∂p/∂T)_V and (∂S/∂p)_T=−(∂V/∂T)_p.

For a fixed amount of simple compressible matter, C_V=(∂U/∂T)_V and C_p=(∂H/∂T)_p.

A cyclic heat engine drawing heat Q_h from a reservoir T_h and rejecting Q_c to T_c delivers W=Q_h−Q_c.

Thermal equilibrium is transitive, permitting a common temperature scale.

For an isolated equilibrium system with fixed energy shell, volume and particle number, the microcanonical model gives equal probability to accessible microstates.

For canonical Z=Σ_i exp(−βE_i), F=−k_BT lnZ, U=−∂_β lnZ and S=k_B(lnZ+βU).

For a system exchanging energy and particles with reservoirs at T and μ, p_i,N=exp[−β(E_i,N−μN)]/Ξ and Ξ=Σ_N,i exp[−β(E_i,N−μN)].

In a classical canonical model, each independent quadratic term in an unrestricted coordinate or momentum contributes k_BT/2 to the mean energy.

For a dilute classical ideal gas in its rest frame, the probability density of a particle velocity is (m/(2πk_BT))³ᐟ²exp[−m|v|²/(2k_BT)].

Pure phases coexist at common T and p when their molar Gibbs energies are equal.

Thermal radiation in a vacuum cavity at temperature T has spectral energy density per angular frequency u_ω=ħω³/[π²c³(exp(ħω/(k_BT))−1)].

For a uniform three-dimensional ideal gas of spinless bosons in the thermodynamic limit, the excited-state capacity reaches n=ζ(3/2)(mk_BT/(2πħ²))³ᐟ² at T_c.

For a uniform dimensionless scalar order parameter m with symmetry m↦−m, an effective free-energy density f=f₀+a m²/2+b m⁴/4 has minima m=0 for a>0 and m²=−a/b for a<0 when b>0.

Connected order-parameter correlations define a correlation length ξ.

Coarse-graining changes couplings while preserving long-distance observables after rescaling.

In the local London model, ∇×j_s=−n_s q_s²B/m_s.

Ginzburg–Landau describes superconductivity through a complex order parameter, symmetry and a gradient/field-dependent free energy, yielding a coherence length and electromagnetic response.